Dixmier trace

In mathematics, the Dixmier trace, introduced by, is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators. Dixmier traces are examples of singular traces.

Some applications of Dixmier traces to noncommutative geometry are described in.

Definition
If H is a Hilbert space, then L1,∞(H) is the space of compact linear operators T on H such that the norm


 * $$\|T\|_{1,\infty} = \sup_N\frac{\sum_{i=1}^N \mu_i(T)}{\log(N)}$$

is finite, where the numbers &mu;i(T) are the eigenvalues of |T| arranged in decreasing order. Let
 * $$a_N = \frac{\sum_{i=1}^N \mu_i(T)}{\log(N)}$$.

The Dixmier trace Tr&omega;(T) of T is defined for positive operators T of L1,∞(H) to be


 * $$\operatorname{Tr}_\omega(T)= \lim_\omega a_N$$

where lim&omega; is a scale-invariant positive "extension" of the usual limit, to all bounded sequences. In other words, it has the following properties:
 * lim&omega;(&alpha;n) ≥ 0 if all &alpha;n ≥ 0 (positivity)
 * lim&omega;(&alpha;n) = lim(&alpha;n) whenever the ordinary limit exists
 * lim&omega;(&alpha;1, &alpha;1, &alpha;2, &alpha;2, &alpha;3, ...) = limω(&alpha;n) (scale invariance)

There are many such extensions (such as a Banach limit of &alpha;1, &alpha;2, &alpha;4, &alpha;8,...) so there are many different Dixmier traces. As the Dixmier trace is linear, it extends by linearity to all operators of L1,∞(H). If the Dixmier trace of an operator is independent of the choice of lim&omega; then the operator is called measurable.

Properties

 * Tr&omega;(T) is linear in T.
 * If T ≥ 0 then Tr&omega;(T) ≥ 0
 * If S is bounded then Trω(ST) = Tr&omega;(TS)
 * Trω(T) does not depend on the choice of inner product on H.
 * Tr&omega;(T) = 0 for all trace class operators T, but there are compact operators for which it is equal to 1.

A trace &phi; is called normal if &phi;(sup xα) = sup &phi;( x&alpha;) for every bounded increasing directed family of positive operators. Any normal trace on $$L^{1,\infty}(H)$$ is equal to the usual trace, so the Dixmier trace is an example of a non-normal trace.

Examples
A compact self-adjoint operator with eigenvalues 1, 1/2, 1/3, ... has Dixmier trace equal to 1.

If the eigenvalues μi of the positive operator T have the property that
 * $$\zeta_T(s)= \operatorname{Tr}(T^s)= \sum{\mu_i^s}$$

converges for Re(s)>1 and extends to a meromorphic function near s=1 with at most a simple pole at s=1, then the Dixmier trace of T is the residue at s=1 (and in particular is independent of the choice of ω).

showed that Wodzicki's noncommutative residue of a pseudodifferential operator on a manifold M of order -dim(M) is equal to its Dixmier trace.